Belief propagation is a well-studied message-passing algorithm that runs over graphical models and can be used for approximate inference and approximation of local marginals. The resulting approximations are equivalent to the Bethe-Peierls approximation of statistical mechanics. Here, we show how this algorithm can be adapted to the world of projected-entangled-pair-state tensor networks and used as an approximate contraction scheme. We further show that the resultant approximation is equivalent to the “mean field” approximation that is used in the simple-update algorithm, thereby showing that the latter is essentially the Bethe-Peierls approximation. This shows that one of the simplest approximate contraction algorithms for tensor networks is equivalent to one of the simplest schemes for approximating marginals in graphical models in general and paves the way for using improvements of belief propagation as tensor networks algorithms.

Ultrafast spectroscopies can access the dynamics of electrons and nuclei at short timescales, shedding light on nonequilibrium phenomena in materials. However, development of accurate calculations to interpret these experiments has lagged behind as widely adopted simulation schemes are limited to subpicosecond timescales or employ simplified interactions lacking quantitative accuracy. Here we show a precise approach to obtain the time-dependent populations of nonequilibrium electrons and atomic vibrations (phonons) up to tens of picoseconds, with a femtosecond time resolution. Combining first-principles electron-phonon and phonon-phonon interactions with a parallel numerical scheme to time-step the coupled electron and phonon Boltzmann equations, our method provides microscopic insight into scattering mechanisms in excited materials. Focusing on graphene as a case study, we demonstrate calculations of ultrafast electron and phonon dynamics, transient optical absorption, structural snapshots, and diffuse x-ray scattering. Our first-principles approach paves the way for quantitative atomistic simulations of ultrafast dynamics in materials.

Strange metals possess transport properties that are markedly different from those of a conventional Fermi liquid. Despite strong similarities in behavior exhibited by distinct families, a consistent description of strange metallic transport and, in particular, its evolution from low to high magnetic field strength $H$, is still lacking. The electron nematic $\mathrm{Fe}{\mathrm{Se}}_{1-x}{\mathrm{S}}_x$ is one such strange metal displaying anomalous $H/T$ scaling in its transverse magnetoresistance as well as a separation of transport and Hall lifetimes at low $H$ beyond its (nematic) quantum critical point at $x_c\sim 0.17$. Here we report a study of the Hall response of $\mathrm{Fe}{\mathrm{Se}}_{1-x}{\mathrm{S}}_x$ across $x_c$ in fields up to 33 T. Upon subtraction of a normal $H$-linear component from the total Hall response (imposed by perfect charge compensation), we find a second component, ascribable to strange metal physics, that grows as $1/T$ upon approach to the quantum critical point. Through this decomposition, we reveal that lifetime separation is indeed driven primarily by the presence of the strange metal component.

We unravel the many-body dynamics of a harmonically trapped impurity colliding with a bosonic medium confined in a double well upon quenching the initially displaced harmonic trap to the center of the double well. We reveal that the emerging correlation dynamics crucially depends on the impurity-medium interaction strength allowing for a classification into different dynamical response regimes. For strong attractive impurity-medium couplings the impurity is bound to the bosonic bath, while for intermediate attractions it undergoes an effective tunneling. In the case of weak attractive or repulsive couplings the impurity penetrates the bosonic bath and performs a dissipative oscillatory motion. Further increasing the impurity-bath repulsion results in the pinning of the impurity between the density peaks of the bosonic medium, a phenomenon that is associated with a strong impurity-medium entanglement. For strong repulsions, the impurity is totally reflected by the bosonic medium. To unravel the underlying microscopic excitation processes accompanying the dynamics, we employ an effective potential picture. We extend our results to the case of two bosonic impurities and demonstrate the existence of a qualitatively similar impurity dynamics.

Ti substituting Ru in ${\mathrm{Sr}}_2\mathrm{Ru}{\mathrm{O}}_4$ in small concentrations induces incommensurate spin density wave order with a wave vector $\mathit{Q}\simeq (2\pi /3,2\pi /3)$ corresponding to the nesting vector of two of three Fermi surface sheets. We consider a microscopic model for these two bands and analyze the correlation effects leading to magnetic order through nonmagnetic Ti doping. For this purpose we use a position-dependent mean-field approximation for the microscopic model and a phenomenological Ginzburg-Landau approach, which both deliver consistent results and allow us to examine the inhomogeneous magnetic order. Spin-orbit coupling additionally leads to spin currents around each impurity, which in combination with the magnetic polarization produce a charge current pattern. This is also discussed within a gauge-field theory in both charge and spin channel. This spin-orbit coupling effect causes an interesting modification of the magnetic structure if currents run through the system. Our findings allow a more detailed analysis of the experimental data for ${\mathrm{Sr}}_2{\mathrm{Ru}}_{1-x}{\mathrm{Ti}}_x{\mathrm{O}}_4$. In particular, we find that the available measurements are consistent with our theoretical predictions.

We report thermopower measurements under hydrostatic pressure on the cuprate superconductor ${\mathrm{La}}_{1.6-x}{\mathrm{Nd}}_{0.4}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ (Nd-LSCO), at low temperature in the normal state accessed by suppressing superconductivity with a magnetic field up to $H=31\mathrm{T}$. Using an ac thermopower measurement technique suitable for high pressure and high field, we track the pressure evolution of the Seebeck coefficient $S$. At ambient pressure and low temperature, $S/T$ in Nd-LSCO was recently found to suddenly increase below the pseudogap critical doping $p^{★}=0.23$, consistent with a drop in carrier density $n$ from $n=1+p$ above $p^{★}$ to $n=p$ below. Under a pressure of 2.0 GPa, we observe that the large $S/T$ value just below $p^{★}$ is suppressed. This confirms a previous pressure study based on electrical resistivity and Hall effect, which found that pressure lowers $p^{★}$, thereby reinforcing the interpretation that this effect is driven by the pressure-induced shift of the van Hove point. It implies that the pseudogap only exists when the Fermi surface is hole-like, which puts strong constraints on theories of the pseudogap phase. We also report thermopower measurements on Nd-LSCO and ${\mathrm{La}}_{1.8-x}{\mathrm{Eu}}_{0.2}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ in the charge density wave phase near $p\sim 1/8$, which reveals a weakening of this phase under pressure.

Photon blockade is the result of the interplay between the quantized nature of light and strong optical nonlinearities, whereby strong photon-photon repulsion prevents a quantum optical system from absorbing multiple photons. We theoretically study a single atom coupled to the light field, described by the resonantly driven Jaynes-Cummings model, in which case the photon blockade breaks down in a second-order phase transition at a critical drive strength. We show that this transition is associated to the spontaneous breaking of an antiunitary $\mathcal{PT}$ symmetry. Within a semiclassical approximation, we calculate the expectation values of observables in the steady state. We then move beyond the semiclassical approximation and approach the critical point from the disordered (blockaded) phase by reducing the Lindblad quantum master equation to a classical rate equation that we solve. The width of the steady-state distribution in Fock space is found to diverge as we approach the critical point with a simple power law, allowing us to calculate the critical scaling of steady-state observables without invoking mean-field theory. We propose a simple physical toy model for biased diffusion in the space of occupation numbers, which captures the universal properties of the steady state. We list several experimental platforms where this phenomenon may be observed.

The differential conductance of metal-insulator-metal devices increases when they are joined with Casimir cavities. An imbalance in injection of hot charge carriers from each side of the insulator is increased with thinner cavities that suppress more quantum vacuum modes. The result is an observed increase in conductance. Additional conductance changes, with insulator thickness and other device parameters, are consistent with an imbalance-induced injection of hot carriers. In addition to the conductance changes, we observe anomalous offsets in the current and voltage. We interpret the conductance changes in terms of a $\mathrm{\Delta }E\mathrm{\Delta }t$ uncertainty-principle-like limit to the injection of hot carriers from zero-point fluctuations.

We investigate how photoinduced topological phase transitions and the magnetic-field-induced quantum Hall effect simultaneously influence the Casimir force between two parallel sheets of staggered two-dimensional (2D) materials of the graphene family. We show that the interplay between these two effects enables on-demand switching of the force between attractive and repulsive regimes while keeping its quantized characteristics. We also show that doping these 2D materials below their first Landau level allows one to probe the photoinduced topology in the Casimir force without the difficulties imposed by a circularly polarized laser. We demonstrate that the magnetic field has a huge impact on the thermal Casimir effect for dissipationless materials, where the quantized aspect of the energy levels leads to a strong repulsion that could be measured even at room temperature.

Tensor networks states allow one to find the low-energy states of local lattice Hamiltonians through variational optimization. Recently, a construction of such states in the continuum was put forward, providing a first step towards the goal of solving quantum field theories (QFTs) variationally. However, the proposed manifold of continuous tensor network states (CTNSs) is difficult to study in full generality, because the expectation values of local observables cannot be computed analytically. In this paper we study a tractable subclass of CTNSs, the Gaussian CTNSs (GCTNSs), and benchmark them on simple quadratic and quartic bosonic QFT Hamiltonians. We show that GCTNSs provide arbitrarily accurate approximations to the ground states of quadratic Hamiltonians and decent estimates for quartic ones at weak coupling. Since they capture the short distance behavior of the theories we consider exactly, GCTNSs even allow one to renormalize away simple divergences variationally. In the end our study makes it plausible that CTNSs are indeed a good manifold to approximate the low-energy states of QFTs.

We quantum simulated the 2D Harper-Hofstadter (HH) lattice model in a highly elongated tube geometry—three sites in circumference—using an atomic Bose-Einstein condensate. In addition to the usual transverse (out-of-plane) magnetic flux, piercing the surface of the tube, we threaded a longitudinal flux ${\mathrm{\Phi }}_{\mathrm{L}}$ down the axis of the tube. This geometry evokes an Aharonov-Bohm interferometer, where noise in ${\mathrm{\Phi }}_{\mathrm{L}}$ would readily decohere the interference present in trajectories encircling the tube. We observe this behavior only when transverse flux is a rational fraction of the flux quantum and remarkably find that for irrational fractions the decoherence is absent. Furthermore, at rational values of transverse flux, we show that the time evolution averaged over the noisy longitudinal flux matches the time evolution at nearby irrational fluxes. Thus, the appealing intuitive picture of an Aharonov-Bohm interferometer is insufficient. Instead, we quantitatively explain our observations by transforming the HH model into a collection of momentum-space Aubry-André models.

We describe the use of tensor networks to numerically determine wave functions of interacting two-dimensional fermionic models in the continuum limit. We use two different tensor network states: one based on the numerical continuum limit of fermionic projected entangled pair states obtained via a tensor network formulation of multigrid and another based on the combination of the fermionic projected entangled pair state with layers of isometric coarse-graining transformations. We first benchmark our approach on the two-dimensional free Fermi gas then proceed to study the two-dimensional interacting Fermi gas with an attractive interaction in the unitary limit, using tensor networks on grids with up to 1000 sites.

We highlight a general theory to engineer arbitrary Hermitian tight-binding lattice models in electrical LC circuits, where the lattice sites are replaced by the electrical nodes, connected to its neighbors and to the ground by capacitors and inductors. In particular, by supplementing each node with $n$ subnodes, where the phases of the current and voltage are the $n$ distinct roots of unity, one can in principle realize arbitrary hopping amplitude between the sites or nodes via the shift capacitor coupling between them. This general principle is then implemented to construct a plethora of topological models in electrical circuits, topolectric circuits, where the robust zero-energy topological boundary modes manifest through a large boundary impedance, when the circuit is tuned to the resonance frequency. The simplicity of our circuit constructions is based on the fact that the existence of the boundary modes relies only on the Clifford algebra of the corresponding Hermitian matrices entering the Hamiltonian and not on their particular representation. This in turn enables us to implement a wide class of topological models through rather simple topolectric circuits with nodes consisting of only two subnodes. We anchor these outcomes from the numerical computation of the on-resonance impedance in circuit realizations of first-order ($m=1$), such as Chern and quantum spin Hall insulators, and second- ($m=2$) and third- ($m=3$) order topological insulators in different dimensions, featuring sharp localization on boundaries of codimensionality $d_c=m$. Finally, we subscribe to the stacked topolectric circuit construction to engineer three-dimensional Weyl, nodal-loop, quadrupolar Dirac, and Weyl semimetals, respectively, displaying surface- and hinge-localized impedance.

We numerically solve the Hubbard model on the Bethe lattice with finite coordination number $z=3$, and determine its zero-temperature phase diagram. For this purpose, we introduce and develop the “variational uniform tree state” (VUTS) algorithm, a tensor-network algorithm which generalizes the variational uniform matrix product state algorithm to tree tensor networks. Our results reveal an antiferromagnetic insulating phase and a paramagnetic metallic phase, separated by a first-order doping-driven metal-insulator transition. We show that the metallic state is a Fermi liquid with coherent quasiparticle excitations for all values of the interaction strength $U$, and we obtain the finite quasiparticle weight $Z$ from the single-particle occupation function of a generalized “momentum” variable. We find that $Z$ decreases with increasing $U$, ultimately saturating to a nonzero, doping-dependent value. Our work demonstrates that tensor-network calculations on tree lattices, and the VUTS algorithm in particular, are a platform for obtaining controlled results for phenomena absent in one dimension, such as Fermi liquids, while avoiding computational difficulties associated with tensor networks in two dimensions. We envision that future studies could observe non-Fermi liquids, interaction-driven metal-insulator transitions, and doped spin liquids using this platform.

The recent investigation of chains of Rydberg atoms has brought back the problem of commensurate-incommensurate transitions into the focus of current research. In two-dimensional classical systems or in one-dimensional quantum systems, the commensurate melting of a period-$p$ phase with $p$ larger than 4 is known to take place through an intermediate floating phase where correlations between domain walls or particles decay only as a power law, but when $p$ is equal to 3 or 4, it has been argued by Huse and Fisher that the transition could also be direct and continuous in a nonconformal chiral universality class with a dynamical exponent larger than 1. This is only possible, however, if the floating phase terminates at a Lifshitz point before reaching the conformal point, a possibility debated since then. Here we argue that this is a generic feature of models where the number of particles is not conserved because the exponent of the floating phase changes along the Pokrovsky-Talapov transition and can, thus, reach the value at which the floating phase becomes unstable. Furthermore, we show numerically that this scenario is realized in an effective model of the period-3 phase of Rydberg chains in which hard-core bosons are created and annihilated three by three: The Luttinger liquid parameter reaches the critical value $p^2/8=9/8$ along the Pokrovsky-Talapov transition, leading to a Lifshitz point that separates the floating phase from a chiral transition. Implications beyond Rydberg atoms are briefly discussed.

We show a way to perform the canonical renormalization group (RG) prescription in tensor space: write down the tensor RG equation, linearize it around a fixed-point tensor, and diagonalize the resulting linearized RG equation to obtain scaling dimensions. The tensor RG methods have had great success in producing accurate free energy compared with the conventional real-space RG schemes. However, the above-mentioned canonical procedure has not been implemented for general tensor-network-based RG schemes. We extend the success of the tensor methods further to extraction of scaling dimensions through the canonical RG prescription, without explicitly using the conformal field theory. This approach is benchmarked in the context of the Ising models in one dimension and two dimensions. Based on a pure RG argument, the proposed method has potential applications to three-dimensional systems, where the existing bread-and-butter method is inapplicable.

Hole gases in planar germanium can have high mobilities in combination with strong spin-orbit interaction and electrically tunable $g$ factors, and are therefore emerging as a promising platform for creating hybrid superconductor-semiconductor devices. A key challenge towards hybrid Ge-based quantum technologies is the design of high-quality interfaces and superconducting contacts that are robust against magnetic fields. In this work, by combining the assets of aluminum, which provides good contact to the Ge, and niobium, which has a significant superconducting gap, we demonstrate highly transparent low-disordered JoFETs with relatively large $I_{\text{C}}{R}_{\text{N}}$ products that are capable of withstanding high magnetic fields. We furthermore demonstrate the ability of phase-biasing individual JoFETs, opening up an avenue to explore topological superconductivity in planar Ge. The persistence of superconductivity in the reported hybrid devices beyond 1.8 T paves the way towards integrating spin qubits and proximity-induced superconductivity on the same chip.

The low-energy band structure of few-layer ${\mathrm{MoS}}_2$ is relevant for a large variety of experiments ranging from optics to electronic transport. Its characterization remains challenging due to complex multiband behavior. We investigate the conduction band of dual-gated three-layer ${\mathrm{MoS}}_2$ by means of magnetotransport experiments. The total carrier density is tuned by voltages applied between ${\mathrm{MoS}}_2$ and both top and bottom gate electrodes. For asymmetrically biased top and bottom gates, electrons accumulate in the layer closest to the positively biased electrode. In this way, the three-layer ${\mathrm{MoS}}_2$ can be tuned to behave electronically like a monolayer. In contrast, applying a positive voltage on both gates leads to the occupation of all three layers. Our analysis of the Shubnikov–de Haas oscillations originating from different bands lets us attribute the corresponding carrier densities in the top and bottom layers. We find a twofold Landau level degeneracy for each band, suggesting that the minima of the conduction band lie at the $\pm K$ points of the first Brillouin zone. This is in contrast to band structure calculations for zero layer asymmetry, which report minima at the $Q$ points. Even though the interlayer tunnel coupling seems to leave the low-energy conduction band unaffected, we observe scattering of electrons between the outermost layers for zero layer asymmetry. The middle layer remains decoupled due to the spin-valley symmetry, which is inverted for neighboring layers. When the bands of the outermost layers are energetically in resonance, interlayer scattering takes place, leading to an enhanced resistance and to magneto-interband oscillations.

Photon number resolving detectors play a central role in quantum optics. A key challenge in resolving the number of absorbed photons in the microwave frequency range is finding a suitable material that provides not only an appropriate band structure for absorbing low-energy photons but also a means of detecting a discrete photoelectron excitation. To this end, we propose to measure the temperature gain after absorbing a photon using superconducting cadmium arsenide (${\mathrm{Cd}}_3{\mathrm{As}}_2$) with a topological semimetallic surface state as the detector. The surface electrons absorb the incoming photons and then transfer the excess energy via heat to the superconducting bulk's phonon modes. The temperature gain can be determined by measuring the change in the zero-bias bulk resistivity, which does not significantly affect the lattice dynamics. Moreover, the obtained temperature gain scales discretely with the number of absorbed photons, enabling a photon-number resolving function. Here, we will calculate the temperature increase as a function of the number and frequency of photons absorbed. We will also derive the timescale for the heat transfer process from the surface electrons to the bulk phonons. We will specifically show that the transfer processes are fast enough to ignore heat dissipation loss.